Uncountable groups and the geometry of inverse limits of coverings

Abstract

In this paper we develop a new approach to the study of uncountable fundamental groups by using Hurewicz fibrations with the unique path-lifting property (lifting spaces for short) as a replacement for covering spaces. In particular, we consider the inverse limit of a sequence of covering spaces of X. It is known that the path-connectivity of the inverse limit can be expressed by means of the derived inverse limit functor lim ← − 1 , which is, however, notoriously difficult to compute when the π1(X) is uncountable. To circumvent this difficulty, we express the set of path-components of the inverse limit, X, in terms of the functors lim ← − and lim ← − 1 applied to sequences of countable groups arising from polyhedral approximations of X.A consequence of our computation is that path-connectedness of lifting space implies that π1( X) supplements π1(X) in π1(X) where π1(X) is the inverse limit of fundamental groups of polyhedral approximations of X. As an application we show that Gwhere F is the canonical inverse limit of finite rank free groups, G is the fundamental group of the Hawaiian Earring, and KerA( F ) is the intersection of kernels of homomorphisms from F to A.